Skip to main content
SpringerLink
Log in

Completion of Linear Differential Systems to Involution

  • Conference paper
Book cover Computer Algebra in Scientific Computing CASC’99

Abstract

In this paper we generalize the involutive methods and algorithms have been devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutive algorithms. We present an algorithm for computation of a minimal involutive differential basis. Its correctness and termination hold for any constructive and noetherian involutive division. As two important applications we consider posing of an initial value problem for a linear differential system providing uniqueness of its solution and Lie symmetry analysis of nonlinear differential equations. In particular, this allows to determine the structure of arbitrariness in general solution of linear systems and thereby to find the size of symmetry group.

This work was supported by grant INTAS-96-184 and grants No.96-15-96030, 98-01-00101 from Russian Foundation for Basic Research.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cartan, E. : Sur certaines expressions différentielles a le problème de Pfaff. Annates Ecole Normale, 3-e serie, 16, 1899, 239–332; Sur l’integration des systèmes d’équations aux différentielles totales. Ibid., 18 (1901) 241–311.

    MathSciNet  Google Scholar 

  2. Kähler, E.: Einführung in die Theorie der Systeme von Differentialgleichungen, Teubner, Leipzig, 1934.

    MATH  Google Scholar 

  3. Cartan, E.: Les Systèmes Differentielles Extérieurs et leurs Applications Géometriques, Paris, Hermann, 1945.

    Google Scholar 

  4. Arais, E.A., Shapeev, V.P., Yanenko, N.N.: Realization of Cartan’s Method of Exterior Differential Forms on an Electronic Computer. Sov. Math. Dokl. 15(1) (1974) 203–205.

    Google Scholar 

  5. Hartley, D., Tucker, R.W.: Constructive Implementation of the Cartan-Kähler Theory of Exterior Differential Systems. J. Symb. Comp. 12 (1991) 655–667.

    Article  MathSciNet  MATH  Google Scholar 

  6. Riquier, C.: Les Systèmes d’Equations aux Dérivées Partielles, Gauthier- Villars, Paris, 1910.

    Google Scholar 

  7. Rust, C.J., Reid G.J.: Rankings of Partial Derivatives. In: Proceedings of IS- SAC’97, W.Kxichlin (ed.), ACM Press, 1997 pp. 9–16.

    Google Scholar 

  8. Janet, M.: Leçons sur les Systèmes d’Equations aux Dérivées Partielles, Cahiers Scientifiques, IV, Gauthier-Villars, Paris, 1929.

    Google Scholar 

  9. Schwarz, F.: The Riquier-Janet Theory and its application to Nonlinear Evo¬lution Equations. Physica 11D (1984) 243–251.

    Google Scholar 

  10. Topunov, V.L.: Reducing Systems of Linear Differential Equations to a Passive Form. Acta Appl Math. 16 (1989) 191–206.

    Article  MathSciNet  MATH  Google Scholar 

  11. Schwarz. F. : An Algorithm for Determining the Size of Symmetry Groups. Computing 49 (1992) 95–115.

    Article  MathSciNet  MATH  Google Scholar 

  12. Pommaret, J.F.: Systems of Partial Differential Equations and Lie Pseudo- groups, Gordon & Breach, New York, 1978.

    Google Scholar 

  13. Pommaret, J.F.: Partial Differential Equations and Group Theory. New Perspectives for Applications, Kluwer, Dordrecht, 1994.

    MATH  Google Scholar 

  14. Schü, J., Seiler, W.M., Calmet, J.: Algorithmic Methods for Lie Pseudogroups. In: Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, N.Ibragimov et al. (eds.), Kluwer, Dordrecht, 1993, pp. 337–344.

    Chapter  Google Scholar 

  15. Seiler, W.M. : Applying AXIOM to Partial Differential Equations. Internal Report 95-17, Universität Karlsruhe, Fakultat Informatik, 1995.

    Google Scholar 

  16. Thomas, J.: Differential Systems. AMS Publication, New York, 1937.

    Google Scholar 

  17. Boulier, F., Lazard, D., Ollivier, F., Petitot M.: Representation for the Radical of a Finitely Generated Differential Ideal. In: Proceedings of ISSAC’95, A.H.M. Levelt (ed.), ACM Press, 1995, pp. 158–166.

    Google Scholar 

  18. Zharkov, A.Yu., Blinkov, Yu.A.: Involutive Approach to Investigating Polynomial Systems. In: Proceedings of “SC 93”, International IMACS Symposium on Symbolic Computation: New Trends and Developments (Lille, June 14–17, 1993). Math. Comp. Simul. 42 (1996) 323–332.

    Article  MathSciNet  Google Scholar 

  19. Gerdt, V.P., Blinkov, Yu.A.: Involutive Bases of Polynomial Ideals. Preprint-Nr. 1/1996, Naturwissenschaftlich-Theoretisches Zentrum, University of Leipzig; Math. Comp. Simul. 45 (1998) 519–542.

    Article  MathSciNet  MATH  Google Scholar 

  20. Apel, J.: Theory of Involutive Divisions and an Application to Hilbert Function. J. Symb. Comp. 25 (1998) 683–704.

    Article  MathSciNet  MATH  Google Scholar 

  21. Gerdt, V.P., Blinkov, Yu.A.: Minimal Involutive Bases. Math. Comp. Simul. 45 (1998) 543–560.

    MathSciNet  MATH  Google Scholar 

  22. Gerdt, V.P., Berth, M., Czichowski, G.: Involutive Divisions in Mathematica: Implementation and Some Applications. In: Proceedings of the 6th Rhein Workshop on Computer Algebra (Sankt-Augustin, Germany, March 31 - April 3, 1998), J. Calmet (Ed.), Institute for Algorithms and Scientific Computing, GMD-SCAI, Sankt-Augustin, 1998, pp. 74–91.

    Google Scholar 

  23. Gerdt, V.P. : Involutive Division Technique: Some Generalizations and Optimizations, Preprint JINR E5–98–151, Dubna, 1998. To be published in the Proceedings of “CASC’98” (April 20–24, 1998, St.Petersburg).

    Google Scholar 

  24. Reid, G.J.: Algorithms for Reducing a System of PDEs to Standard Form, Determining the Dimension of its Solution Space and Calculating its Taylor Series Solution. Euro. J. Appl. Maths. 2 (1991) 293–318.

    MATH  Google Scholar 

  25. Reid, G.J., Wittkopf, A.D., Boulton A.: Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Form. Euro. J. Appl. Maths. 7 (1996) 635–666.

    MathSciNet  MATH  Google Scholar 

  26. Carra’Ferro, G. : Grobner Bases and Differential Algebra. Lec. Not. in Comp. Sci. 356 (1987) 129–140.

    Article  MathSciNet  Google Scholar 

  27. Ollivier, F.: Standard Bases of Differential Ideals. Lec. Not. in Comp. Sci. 508 (1990) 304–321.

    Article  MathSciNet  Google Scholar 

  28. Hereman, W. : Symbolic software for the computation of Lie symmetry analysis. In: CRC Handbook of Lie Group Analysis of Differential Equations, Volume 3: New Trends in Theoretical Developments and Computational methods, Ibragimov, N.H. et al. (eds.), CRC Press, Boca Raton, 1995, pp. 367–413.

    Google Scholar 

  29. Reid, G.J.: Finding Abstract Lie Symmetry Algebras of Differential Equations without Integrating Determaning Equations. Euro. J. Appl. Maths. 2 (1991) 319–340.

    Article  MATH  Google Scholar 

  30. Mansfield, E., Clarkson, P.A.: Application of the Differential Algebra Package diff grob2 to Classical Symmetries of Differential Equations, J. Symb. Comp. 23 (1997) 517–533.

    Article  MathSciNet  Google Scholar 

  31. Ritt, J.F.: Differential Algebra, AMS Publication, New York, 1950.

    MATH  Google Scholar 

  32. Kolchin, E.R.: Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.

    MATH  Google Scholar 

  33. Gerdt, V.P. : Gröbner Bases and Involutive Methods for Algebraic and Differential Equations, Math. Comp. Model. 25, No.8/9 (1997) 75–90.

    Article  MathSciNet  MATH  Google Scholar 

  34. Gerdt, V.P., Berth, M., Czichowski, G., Kornyak V.V.: Construction of Involutive Monomial Sets for Different Involutive Divisions. This volume.

    Google Scholar 

  35. Buchberger, B.: Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. In: Recent Trends in Multidimensional System Theory, Bose, N.K. (ed.), Reidel, Dordrecht, 1985. pp. 184–232.

    Chapter  Google Scholar 

  36. Cox, D., Little, J., O’Shea D.: Ideals, Varieties and Algorithms, 2nd Edition, Springer-Verlag, New York, 1996.

    MATH  Google Scholar 

  37. Lewy. H. : An Example of a Smooth Linear Partial Differential Equation without Solution, Ann. Math 66(1957) 155–158.

    Article  MathSciNet  Google Scholar 

  38. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics 149, Cambridge University Press, Cambridge, UK, 1991.

    Book  Google Scholar 

  39. Bublik, V.V.: Group Classification of Equations for Dynamics of Viscous Heat Conducting Gas, In: Dynamics of Continuous Medium 113, Novosibirsk, 1998, pp. 19–21 (in Russian).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Computing Techniques and Automation, Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Vladimir P. Gerdt

Authors
  1. Vladimir P. Gerdt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Technische Universität, D-80290, München, Germany

    Victor G. Ganzha

  2. Institut für Informatik, Technische Universität, D-80290, München, Germany

    Ernst W. Mayr

  3. Institute of Theoretical and Applied Mathematics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gerdt, V.P. (1999). Completion of Linear Differential Systems to Involution. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing CASC’99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60218-4_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-60218-4_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66047-7

  • Online ISBN: 978-3-642-60218-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation